Show that $\mu_n(\mathbb F_q)=\mu_{\gcd(n,q-1)}(\mathbb F_q)$.
Where $\mu_n(K)=\{ a \in K^\times : a^n=1 \}$.
My thoughts about it, if $a\in \mu_n(K)$ then $a^n=1$ a not equals zero. I need to some how to show that $\gcd(n,q-1)$ also satisfies this relation.
If I write it down as $d=\gcd(n,q-1)=cn+b(q-1)$ then $a^d = a^{b(q-1)}=1$, does this finish the proof? is it even correct?